Revival: Block Method for Solving the Laplace Equation and for Constructing Conformal Mappings (1994)

238 Pages·2017·4.904 MB·English

by  Volkov, Evgenii A

E l. Volkov
Bloch method
?or Solviog the
Laplace Equation
and for Constructing
Conformal mappings
First published 1994 by CRC Press
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iii
Preface
This book is the result of my many years of investigations car­
ried out in the field of approximate methods of solving differential
equations. Chapter 1 contains the exposition and substantiation of
my own block method of solving a mixed boundary-value problem, in­
cluding the Dirichlet problem and the Neumann problem for Laplace’s
equation on an arbitrary polygon. The boundary values of the re­
quired solution or of its normal derivative are defined on the sides of
the polygon by analytic functions, in particular, by algebraic poly­
nomials. An approximate solution of the boundary-value problem is
sought on a finite number of fixed intersecting subdomains (sectors,
half-disks, disks), called basic blocks, in the form of harmonic func­
tions which are elementary under the boundary conditions specified
by algebraic polynomials.
The block method is constructed with the use of integral rep­
resentations of harmonic functions of the type of Poisson’s integral
on extended blocks which are sectors, half-disks and disks (contain­
ing the basic blocks). In these representations the corresponding
integrals are approximated by the quadrature formulas of rectangles.
The resulting system of algebraic linear equations is well conditioned
and its solution can be found by means of simple iterations which
converge in a geometric progression with a common ratio indepen­
dent of the discreteness parameter. The approximate solution of a
boundary-value problem, found by means of the block method, is
exponentially convergent (on the basic blocks) with respect to the
discreteness parameter up to the boundary of the polygon even if
the defined boundary values of the solution are discontinuous at the
vertices of the polygon.
IV Preface
In Chapter 2, the problem of approximate conformal mapping of
multiply-connected polygons onto canonical domains (onto a plane
with cuts along parallel segments and onto a ring with cuts along
the arcs of concentric circles) is constructively solved with the use
of the block method by means of a reduction to the Dirichlet prob­
lem. No preliminary information is required concerning the unknown
conformal invariants of the domains being mapped. The conformal
invariants are determined in the process of constructing of the approx­
imate conformal mapping with an error decreasing exponentially with
respect to the discreteness parameter. The approximate conformal
mapping is obtained on the (basic) blocks in the form of elementary
analytic functions.
Chapter 3 is devoted to the development and application of the
block method to the approximate conformal mapping of simply-con­
nected domains onto a strip and of doubly-connected domains onto
a ring. The algorithms of constructing approximate conformal map­
pings of a number of specific domains are described in detail. The
class of domains being mapped is considerably extended and the ap­
paratus of the block method is developed by means of introduction
of special blocks, the use of analytic continuation, and change of a
complex variable. Very accurate results of calculations carried out
on the BESM-6 computer according to the indicated algorithms are
given which substantiate by practical computations the exponential
convergence of the block method and its strong stability as concern­
ing the rounding-off errors. All the computations were carried out by
me.
In Chapter 4 the block method is applied for constructing approx­
imate conformal mappings of some infinite domains with periodical
structure.
Some references are made to my papers [56-70] concerning the
block method, the contents of papers [59, 67] are partially reflected.
Papers [71, 72] in which the use of the block method for Dirichlet’s
problem with nonanalytical boundary conditions is developed and
the rapid approximate method of constructing Green’s function of
Laplace’s operator on polygons is exposed, are not used in the book.
The prototype of the block method is the method of compos­
ite nets for solving boundary-value problems for Laplace’s equation
Preface V
on domains with piecewise-smooth boundary (see [51-54]). In the
method of composite nets, a polar net convergent at the vertex is
constructed in order to increase accuracy, in the neighborhood of the
vertices of the angles in which the derivatives of the required solutions
have singularities. In the block method, by analogy, a block-sector
on which the corresponding integral representation of the solution of
the boundary-value problem is approximated with the account of the
singularities of the derivatives of all orders is defined in the vicin­
ity of each vertex. Paper [55] served for the transition from the
method of composite nets to the block method. In [55], an approxi­
mate analytic representation of a solution via some quasipolynomials
(functions of singularities) with unknown coefficients whose order is
consistent with the step of the net on the rest part of the domain is
used in the vicinity of the vertices of the angles.
In this book I speak only of the exponentially convergent block
method and do not touch upon other approximate methods of
solving boundary-value problems for Laplace’s equation and con­
structing conformal mappings of simply-connected and multiply-
connected domains. The following monographs are devoted to
other approximate methods of solving these problems: M. A. Alek-
sidze [2], J.-P. Aubin [3]; K. I. Babenko [4]; I. Babuska, M. Prager
and E. Vitasek [5]; N. S. Bakhvalov, N. P. Zhidkov and G. M.
Kobel’kov [6]; I. S. Berezin and N. P. Zhidkov [7]; L. Collatz [9];
E. G. D’yakonov [10,11]; P. F. Fil’chakov [12]; V. P. Fil’chakova [13];
G. Forsythe and W. Wasow [14]; D. Gaier [15]; S. K. Godunov
and V. S. Ryabenkii [16]; L. V. Kantorovich and V. I. Krylov
[19]; W. Koppenfels and F. Stallmann [22]; G. I. Marchuk and
V. I. Agoskov [25]; G. I. Marchuk and Yu. A. Kuznetsov [26];
G. I. Marchuk and V. I. Lebedev [27]; G. I. Marchuk and
V. V. Saidurov [28]; S. G. Mikhlin [30, 31]; A. Mitchell and
R. Wait [32]; L. A. Oganesyan and L. A. Rukhovets [33];
V. S. Ryabenkii [36]; V. S. Ryabenkii and A. F. Filippov [37];
A. A. Samarskii [38]; A. A. Samarskii and V. B. Andreev [39];
A. A. Samarskii and A. V. Gulin [40]; A. A. Samarskii, R. D. Lazarov,
and V. L. Makarov [41]; A. A. Samarskii and E. S. Nikolaev [42];
V. K. Saulyev [43]; G. Strang and G. Fix [46]; V. I. Vlasov [47].
VI
The Author
Evgenii A. Volkov, Ph.D., D.Sc. (Phys.-Math.), is a Lead­
ing Researcher at Steklov Mathematical Institute of the Russian Aca­
demy of Sciences. He was born in 1926 in Tula, Russia. In 1943-1946
he was a Designer at a Munition Factory, Petropavlovsk, U.S.S.R. In
1946-1951 as a student of the Mechanical-Mathematics Department
of Moscow State University he fulfilled a number of original investi­
gations in the field of computing devices. Since 1951 he has been a
Researcher at the Academy of Sciences, Moscow.
Dr. Volkov received his Ph.D. and D.Sc. degrees at Steklov Math­
ematical Institute of the U.S.S.R. Academy of Sciences in 1954 and
1967 respectively. From 1958 to 1970 as a Consultant at a research
institute, Dr. Volkov took an active part in the realization of State
Programs. In 1954-1958 he was an Associate Professor at the Moscow
Physico-Technical Institute, and in 1971-1980 he was Professor at the
Moscow Institute of Physical Engineering. Currently he is the Assis­
tant Editor-in-Chief of the Proceedings of the Steklov Mathematical
Institute.
Dr. Volkov fulfills scientific investigations in the field of applied
mathematics (numerical methods for solving ordinary and partial dif­
ferential equations, some nonlinear problems, etc.) from 1949, and
possesses great experience in using high-speed computers from 1952.
He published over 100 scientific articles in which he combined novelty,
strict mathematical substantiation, and practical trends. Dr. Volkov
is the author of the book Numerical Methods, Nauka Publ., Moscow,
1982, which has had two editions in Russian and was translated into
English and Spanish.
vii
Contents
Chapter 1. Approximate Block Method for Solving the
Laplace Equation on Polygons......................................... 1
1. Setting up a Mixed Boundary-Value Problem for the
Laplace Equation on a Polygon.......................................... 1
1.1. Description of the Boundary of a Polygon............... 1
1.2. Setting up a Boundary-Value Problem ................ 7
2. A Finite Covering of a Polygon by Blocks of Three
Types.................................................................................... 13
3. Representation of the Solution of a Boundary-Value
Problem on Blocks.............................................................. 19
3.1. Carrier Functions..................................................... 19
3.2. Preliminary Lemmas............................................... 24
3.3. Representation of the Solution onB locks.............. 30
3.4. Additional Lemmas.................................................. 32
4. An Algebraic Problem........................................................ 34
5. The Main Result................................................................. 40
5.1. Theorem on the Convergence of the Block
Method..................................................-.................... 40
5.2. Discussion of the Theorem..................................... 42
6. Proofs of Theorem 5.1 and of Lemmas 4.1-4.4.............. 48
6.1. Preliminary Lemmas............................................... 48
6.2. Theorem on the Solvability of an Algebraic
Problem.................................................................... 54
6.3. Fundamental Lemmas............................................... 58
Vlll Contents
6.4. Proof of Theorem 5.1............................................... 66
6.5. Proofs of Lemmas 4.2-4.4........................................ 67
7. The Stability and the Labor Content of Computations
Required by the Block Method........................................ 73
7.1. Preliminary Lemmas ............................................... 73
7.2. Verification of the Criterion of Solvability of an
Algebraic Problem..................................................... 77
7.3. The Stability of the Block Method in a Uniform
M etric....................................................................... 79
7.4. The Labor of Computations Required by the Block
Method....................................................................... 83
8. Approximation of a Conjugate Harmonic Function on
Blocks.................................................................................... 85
9. Neumann’s Problem........................................................... 92
9.1. The Solvability of Neumann’s Problem on a
Polygon....................................................................... 92
9.2. Approximate Solving of Neumann’s Problem by
Block Method........................................................... 96
10. The Case of Arbitrary Analytic Mixed Boundary
Conditions.......................................................................... 98
Chapter 2. Approximate Block Method of Conformal
Mapping of Polygons onto Canonical Domains . . . . 108
11. Approximate Conformal Mapping of a Simply- Connected
Polygon onto a Disk..............................................................108
12. Basic Harmonic Functions ..................................................Ill
13. Approximate Conformal Mapping of a Multiply-
Connected Polygon onto a Plane with Cuts along
Parallel Line Segments ........................................................113
13.1. Structure of a Mapping...............................................113
13.2. Constructing an Approximate Mapping...................115
14. Approximate Conformal Mapping of a Multiply-
Connected Polygon onto a Ring with Cuts along the
Arcs of Concentric Circles.....................................................119
14.1. Structure of a Mapping...............................................119
14.2. Constructing an Approximate Mapping...................122
Contents IX
Chapter 3. Development and Application of the
Approximate Block Method for Conformal
Mapping of Simply-Connected and Doubly-Connected
Domains.....................................................................................124
15. Approximate Conformal Mapping of Some Polygons
onto a Strip.............................................................................124
15.1. Scheme of Constructing a Mapping.........................124
15.2. Mapping of a Rectangle..............................................125
15.3. Mapping of Two Octagons onto a Strip and of a
T-shaped Domain onto a Half-Strip.........................133
16. Scheme of Constructing a Conformal Mapping of a
Doubly-Connected Domain onto a Ring............................141
17. Mapping a Square Frame onto a Ring...............................143
18. Mapping a Square with a Circular Hole Using Circular
Lune Block.............................................................................149
19. Representation of a Harmonic Function on a Ring . . . 155
20. Using a Block-Ring for Mapping Domain (18.1) onto a
Ring..........................................................................................157
21. A Block-Bridge.......................................................................161
22. Limit Cases.............................................................................167
23. Mapping a Disk with an Elliptic Hole or with a
Retrosection onto a R ing....................................................169
24. Mapping a Disk with a Regular Polygonal Hole................173
25. Mapping the Exterior of a Parabola with a Hole onto a
Ring.........................................................................................181
Chapter 4. Approximate Conformal Mapping of
Domains with a Periodic Structure by the Block
Method.........................................................................................188
26. Mapping a Domain of the Type of Half-Plane with a
Periodic Structure onto a Half-Plane..................................188
26.1. Construction of a Conformal Mapping of a
Domain of the Type of Half-Plane with a
Periodic Boundary onto a Half-Plane......................188
26.2. Constructing Blocks.....................................................191
26.3. The Algorithm..............................................................194